Optimal L p -discrepancy bounds for second order digital sequences

Abstract

The L p -discrepancy is a quantitative measure for the irregularity of distribution modulo one of infinite sequences. In 1986 Proinov proved for all p > 1 a lower bound for the L p -discrepancy of general infinite sequences in the ddimensional unit cube, but it remained an open question whether this lower bound is best possible in the order of magnitude until recently. In 2014 Dick and Pillichshammer gave a first construction of an infinite sequence whose order of L 2 -discrepancy matches the lower bound of Proinov. Here we give a complete solution to this problem for all finite p > 1. We consider socalled order 2 digital (t, d)-sequences over the finite field with two elements and show that such sequences achieve the optimal order of L p -discrepancy simultaneously for all p ∈ (1, ∞).

Discrepancy of Second Order Digital Sequences in Function Spaces With Dominating Mixed Smoothness

“…It is also known that the lower bound (3) for infinite sequences is best possible for all p ∈ (1, ∞). This was shown by the authors of the present paper in the recent work [13] where explicit constructions S d of infinite sequences in arbitrary dimensions d were provided whose L p -discrepancy satisfies…”

“…The conceptual difference in the concept between the discrepancy of finite point sets and infinite sequences is pointed out in [27] and [13]: for finite point sets one is interested in the behavior of the whole set {x 0 , x 1 , . .…”

“…The sequences and methods developed in [13] are the fundamental basis for the results that will be presented in this paper (see, for example, Lemma 4.5).…”

4. Recent advances in higher order quasi-Monte Carlo methods

“…Indeed, recent papers [42,15,16,4] use the system of Haar functions and succeed in generalizing or extending the result of [25]. For instance, it was proven in [15] that order 2 (instead of order 5) digital sequences achieve the best possible order of L 2 -discrepancy. Also, prior to the works of [36,37,38], Hinrichs et al used the system of Faber functions to analyze the worst-case error of order 2 digital nets for different function spaces [40].…”

“…sequences). More recently, several refined analyses for generalizing or extending the work of Dick and Pillichshammer have been conducted [10,42,15,16,4].…”